The potential energy surfaces (PES) of 28 simple rare-gas (Rg)-linear molecule van der Waals (vdW) complexes were calculated using four dispersion-corrected density functionals (HFPBE-XDM, PW86PBE-XDM, a PW86PBE-XDM hybrid functional, and B97-D) and compared with accurate coupled cluster CCSD(T) reference data. In this benchmark study, the quality of the PES are assessed based on the values of the binding energies (BEs), and the geometry at the global minimum and the overall anisotropies and shapes of the PES. All functionals perform adequately on Rg-H₂, Rg-N₂, Rg-CO, Rg-OCS, and Rg-CO₂ dimers. The functionals display various problems for the remaining vdW molecules. In particular, B97-D predicts overbound global minima and incorrect geometries for He,Ne,Ar,Kr-HCN and Ne,Ar,Kr-C₂H₂, PW86PBE-XDM and the hybrid functional massively overbind Rg-HF, and HFPBE-XDM predicts different global minima than the reference PES for Rg-HCl and Rg-HBr complexes. The error trends of the functionals are assessed relative to the size of the Rg atoms and the polarity of the linear molecules. Based on this assessment, we find that delocalization error affects the performance of the functionals and depends on the polarity of the linear molecule. The hybrid functional, which we denote as PW1PBE-XDM, provides lower error statistics than other functionals, which leads to a possibility that this functional can be applied to study other classes of vdW molecules in future PES studies.